a.  of  1. 

DEPARTrOEOT  of  AR-CUITCCTORE 


SWAbtS  And  SWAb0ai5 


::.% 
"i'^ 


4 


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SHALES      ..FD      SH<^DOrs 
Notes   Arr^.n^cd  For 
Tlic  Dep-.rt-vor.t    of  Arcliitcctr.re 

UNIl^RSXTr     OF      ILLINOIS  — 


Mimoegr?cphecl  by 

Student  Supply  Store 
Ghr.mpc^.ign,    ill. 


1 


1. 

•,  rIOTvf'  M.d:Mi(\^  m     :^aLPK  PAHNING 

1.  IlITRODUCTION; 

Objects  are  QYi-it»le  to  the '  eye  owing  to  the  re- 
flection from  their  surface 'ro.ys  of  light,  These  reflected  rays 
strike  upon  the  retina  of  the  eye  and  give  the  sensation  of  sight. 
The  greatest  source  of  light  is  the  sun,  and  rays  from  it  strike 
all  bodies  on  the  earth  at  various  angles.  If  the  angles  of  impact 
of  all  the  rays  efriking  on  the.  surface  be  ^:- the  .'same,  the  surface 
appears  flat,  technically ' "plane" .  If  the  angles  of  impact  are 
different  then  the  surface  appears  "curved". 

Since  therefore,  the  pleasant  or  unpleasant  effect 
of  any  object  v.'ill  depend  upon  the  producing  of  an  effect  of 
pleasing  or  unpleasing  lighting  upon  the  eye,  it  becomes  of 
primary  importance  that  the  students  of  those  branches  of  art 
that  deal  xjxVa   objects  in  the  round,  vjith  objects  subject  for 
their  effect  upon  the  play  of  light  and  shade  upon  thail'  surface, 
namely,  Architecture  and  Sculpture,  be  thoroughly  familiar  vjith 
the  phenomena  of  light  and  shade. 

In  addition  to  this  it  is  essential  that  he  v;ho;  - 
conceives  an  idea  m.ust  be  able  to  express  that  idea  forcibly  and 
convincingly  to  others,  else  his  idea  becomes  of  no  importance. 
The  painter  expresses  an  idea  in  one  vjay,  the  -vTritcr  in  another, 
the  sculptor  in  another,  the  architect  in  still  another.  The 
Sculptor  v7orks.  out  his  idea  in  clay,  then  piaster,  then  in  stone 
or  in  bronze;  the  architect  has  certain  conditions  set  before 
him;  he  v/orks  '  out  the  idea  in  the  drafting  room  and  working  under 
his  direction,  the  builder  oxcc^^tes  that  idea  in  permanent 
materials  of  wood,  brick,  and  stone. 

A  building  docs  not  consist  merely  of  the  lines  by 
which  it  is  represented  in  geometrical  drawing,  but  of  masses 
and  these  are  better  and  more  quickl"  represented  by  tints  than 
by  mere  line  drawing  and  in  order  that  the  final  results  may  be 
that  pleasing  creation  that  distinguishes  the  thing  of  art  from 
that  of  pure  utility-that  make  of  it  architcctmre-the  architect 
must  be  thoroughly  familiar  with  those  feb:if:gal  phenomena  which 
will  make  of  his  creation  a  thing  of  beauty  and  must  be  able  to 
represent  those  phenomena  on  drav/ings . 

In  order  that  a  representation  of  light  and  shade 
may  be  made  upon  geometrical  drawings,  the  architect  brings  into 
use  an  application  of  certain  principles  of  descriptive  geometry, 
using  that  science,  however,  not  as  a  means  alone,  but  as  a  means 
toward  an  end,  and  baaing  the  general  underlying:  principles  of 
the  application  upon  a  careful  observ:^nceof  natural  phenomena- 
adopting  convenient  conventions--never  violating  the  fundamental 


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2. 

§?;'^^  AIIT   'v_^U)OVS, 
principles    of  "n.-^.turr.l    -;-  ?r-Oi.iena. 

2.  .LIGHT_RAYS: 

The   ray®  of  light  ivhich  .-^.re  deo.lt  v/ith  in  reference 
to  anyr-- one  body'rxre  first-direct  or  incident  rays,  second- 
tangential  t?ays, -third-lateral  rays.  Direct  rays  impinge  directly 
upon  the  body.  Tangential  raye  are  those  tangent  to  its  surface. 
Lateral  rays  do. .not  strike  the  surface,  "but  go  on  to  illuminate 
bodies  beyond,  "^'aturally,  in  the  study  of  Sh^ides  "  and  Shadovjs , 
direct  and  tangential  rays  are  those  of  im.portance.   Fig.  1 

3.  SHADi:   ArlD   SHADOWS  ; 

If  rays  of  light  are  excluded  from  certain  portions 
of  a  surface  by  the  shape  of  that  surface  itself,  then  the  dark- 
ened portion  is  said  to  be  in  shade,  if  the  surface,  by  projection 
beyond  or  position  betv^een  the  source  of  light  and  another  surface, 
exclude  light  from  any  portion  of  the  second  surface,  then  the 
first  on?  is  said  to"cast  shadovr  over  the  second".  Shadovrs  reveal 
by  their' extent  the  relative  position  of  as  '.'jell  as  the  shape  of 
surfaces.  Shade  reveals  merely  the  shape  of  separate  surfaces. 

4.  KE/iSOIIS    FOR   AS_SU?^?TI_Oj£  OF 

c:^KTA  III  "cokv-':jtttous  . ' 

Any  object  placed  in  a  fixed  position  on  the  earth, 
as  is  of  course  every  building,  vTill  be  subject  to  a  continual 
change  of  lighting,  to  a  continually  changing  play  of  light,  shade 
and  shadow.  If  on  the  geometrical  or  line  drav;ing  of  an  element 
architectural,  each  separate  artist  ;7ere  to  assume  a  direction 
for  the  rays  of  light  that  are  presuj-tied  to  illui'iinate  an  object-- 
for  naturally  an  assumption  of  directions  of  light  is  prerequisite 
to  the  Tv'orking  out  of  shades  and  shado"'fs  geometrically-then  to 
each'drav/ing  y:ould  have  to  be  added  a  statement  of  the  assumptions 
made.  The  assumption  of  certain  directions  of  the  rays  v^rould  be 
cause  of  great  difficulties  in" the  geometiical  solution  and  a 
geheral  confusion  v/ouia  result,  though  each  distinct  method  or 
means  'ivould  itself  be  correct.  Since  the  finished  result  of  the 
artist's  efforts  must  be  easily  and  clearly  readable  by  those 
who  do  not  understand  the  methods  used  in  acheiving  the  result 
set  before  them;  since  the  v/orking  out  of  shadou's  is  in  any  event 
a  long  and  complicated  process,  and  since,  therefore,  the  adoption 
of  any  conventions  tending  to  make  for  the  dra.ftsman  greater  else 
of  v/orking,  are  of  prime  importance,  and  for  these  reasons  solely 


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5. 


SHADES  Mi'D   SHADDWS 

cert''.in  conventions  as  to  direction  of  light  have  been  universally 
adopted,  Tiie  student, hovfever,  must  not  come  to  believe  that  the 
adoption  for  sake  of  convehience  of  particular  conventions  affect 
in  any  ivay  the  general  principles  underlying  the  Study  of  Shades 
and  Shadows. 

5.  THE  RAY  A^  45fo 

In  the  study,  therefore,  of  Shades  and  Shadovrs,  the  sun  is 
assumed  to  be  the  source  of  light,  and  the  rays  assumed  to  take 
a  dov;nv;ard  direction,  and  to  the  right  parr-^allel  to  the  diagonal 
of  a  cube  (Fig.  2  ),  The  angle  which  the  ray  makes  vjith  its  owa 
projection  .is,  therefore,  35%  15'  52'',  and  is  Icnovm  visually  as 
the  angle  ;|:.  or  the"true  angle  of  the  ray". 

The  projections  of  this  one  ray  are  three,  and  each 
projection  naturally  raakes  an  angle  of  ^5%   v/ith  the  horizontal, 
and  since  they  originate  at  inf inity--the  sun--the  pi'ojections 
of  all  rays  In  any  particular  plane  are  parallel.  (Fig, 3) 
Expressed  architecturally,  there  is  one  ray  vith  a  front ' elevatiQn, 
a  side  elevation  and  a  plan  as  can  be  readily  understood.  The  use 
of  the  ray  iii  this  particular  dilr-ection,  gives  not  only  ease  of 
construction,  but  also  ease  of  interpretation  of  projections  of 
surfaces  on  beyond  the  other,  for  naturally  the  v/idth  of  the 
shador;  cast  by  one  plane  surface  over  another  v/ill  be  exactly 
equal  to  the  projection  of  the  first  surface  in  front  of  the 
second.  Having  established  a  definite  basis  for  study  there  can 
no7'  be  discussed  the  various  procedures  necessar^'  for  the  actual 
finding  of  Shades  and  Shadov;s  under  any  given  condition.  To  the 
study  the  architectural  student  must  bring  more  than  a  mere 
laaoTJledge  of  the  processes  of  descriptive  geometry.  He  must  bring 
a  sense  of  analysis  and  thoughtfulness  that  will  enable  him  to 
discover  in  each  problem  those  particular  elements  that  are 
necessary  for  the  rapid  solution  of  the  problem,  but  he  must 
above  all  else  cultivate  a  knovvledge  of  the  general  shapes  of 
actual  Shades  and  Shadov;s  and  a  common  sense  way  of  looking  at  ■ 
any  problem  presented. 

By  a  m.ere  application  of  principles  of  descriptive  geometry 
the  finding  of  the  precise  piercing  point  of  a  line  on  a  surface- 
any  profile  in  shades  and  shadows  can  be  solved,  but  since 
rapidity  of  thought  ahd  action  as  well  as  accuracy  of  result  are 
the  prime  requisites  of  the  architectural  draftsman  every  effort 
must  be  made  to  learn  the  shortest  and  the  easiest  methods  of 
acheiving  a  given  result.  Analysis  and  acquired  experience, 
thoughfulness ,  Imagination,  observation  of  natural  phenomena, 
these  all  will  give  rapidity  and  accuracy. 


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4. 

THE    FINDING   OK   GEQIiIETRICAL  DKAVflNGS 
OP   SH/.DES    AND   S'rl/DOWS 


BY  GEO'ITITRICAL  MEANS 


6.  DEFINITIONS, 


"A  plane  of  Rays"  is  a  plane  -A'hich  raay  be  considered  as 
made  up  of  the  rays  passing  through  adjacent  points  of  a  straight 
line,  (Fig,  4) 

"Point  of  loss"-The  point  of  intersection  of  a  shadov/  v/ith 
a  shade  line,  or  of  a  shadovf  or  shade  line  virith  the  line  of 
division  betv/een  a  lighted  and  unlighted  sr.rface.  (Fig.  5) 

"Invisible  shddorrs"  or  shadoivs  in  space" that  portion 

of  space  from  iThixih  light  is  excluded  by  a  body  in  direct  light. 

(Pig.  5) 

7.  I'iET'riODS  OP  CONSTRUCTION. 


The  finding  of  sbadovjs  comprises  tv/o  distinnt  operations. 
They  are  in  order  of  consti-^uction:  ^ 

1.  Ti.ie  finding  on  the  object  itself  of  the  line 
v;hich  separates  the  lighted  part  from  that 
shaded  part,  known  in  consequence  as  the 
separation  or  separatrix  or  "shade  line", 

2.  The  finding  of  the  outlines  of  the  s^adov;s 
cast  by  the  object  on  a  foreign  surface,  that 
surface  being  usually  in  the  hoticontal  or 
vertical  plane,  knovm  as  the  "shadow'' , 

Since  it  has  been  assumed  that  the  light  rays  fall  in 
a  definite  position  oblique  to  the  vertical  and  horizontal  plane 
of  the  projeotion  and  since  architectural  details  are  made  up 
for  the  most  part  of  regular  surfaces,  either  planes  at  right  angler 
to  each  other  and  parallel  to  or  at  right  angles  to  the  general ■ 
surfaces  of  revolution  v/hose  elements  are  parallel  or  perpendicular 
to  the  same  plane,  when  it  is  evident  that  in  general  the  shadow 
of  any  object  over  another  object  in  an  6blique  projection  of 
the  first  obgdot,iand  that,  as  can  be  readily  seen,  tHe  outline 
2f   the  first  shadov;  is  the  shadow  of  the  shade  line  of  the  object 
cashing  the  shadow."' 


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5. 


SHADES  AilD  SH/>DOWS 
8.  THE  SHADOWS  £>E  POINTS. 


The  shadow  of  a  point  on  any  surface  is  found  by 
passing  a  ray  (straight  line)  through  that  point  and  finding  v/here 
that  ray  pierces  the  surface.  The  point  of  piercing  must 
necessarily  be  at  the  shado?/  of  the  point..  (Fig.  7) 

9.  THE  SH/iDOW  OF  LIIIES. 

A  straight  line  is  by  definition  made  up  of  a  number  of 
points,  St)  that  the  pas#ing  of  separate  rays  through  each  of 
the  points  v;ould  make  a  plane  of  rays,  the  intersection  of 
which  plane  v/ith  the  surfaces  in  question  would  give  the  line 
of  shadow  of  the  line  considered.  The  length  of  the  shadov;  would 
be  determined  by  the  shadows  of  the  two  points  at  the  ends  of 
the  line  or  by  the  limits  of  the  surfaces  on  which  the  shadow 
falls  (Fig.  8). 

10.  THE  sh;.dows  of  surp;.ces. 


Since  ■^.   surface  is  limited  by  lines,  the  shadow  of  a 
surface  can  evidently  be  found  by  finding  the  shadows  of 
boundary  lines  of  that  surface. 

11.  THE  SK.;D0WS  of  SOLIDS. 


a)  Polyhedra. 

^^re  solids  bounded  by  portions  of  intersecting 
planes.  The  lines  of  demarcation  between  light 
and  shade  viii  natur^.lly  be  along  lines  of 
intersection.  The  outlines  of  the  shado'ws  of 
a  polyhedron  viill  be  determined  by  finding 
the  shadows  of  those  lines  of  intersection  that 
divide  the  lighted  from  the  unlighted  pofction 
of  the  polyhedra  (Fig,  9). 

b)  Surfaces  of  Revolution. 

The  shade  lines  on  surfaces  of  revolution  are 
formed  by  rays  tangent  to  the  surfaces.  The 
ovitline  of  the  shadov?  v;ill  evidently  depend 
upon  the  finding  of  the  shadow  of  that  line 
of  tangency.  (Fig.  10). 


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6. 


SHADES  AND  SHADOWS 

12.  THE  GEIIERAL  PKOBLEli 

Prom  the  preceding  discussions  it  should  be  evident  that 
the  general  problem  involved  is  that  of  ''representing  the  rays 
which  pass  through  points  in  the  shade  line  of  an  object  r-.nd 
finding  the  points  at  which  these  rays  strike  another  object. 
Generally  spealcing,  this  is  not  a  difficult  problem  in 
descriptive  geometry,  and  is  one  quite  v/ithin  the  power  of  an 
architectural  draftsman  of  a  little  experience,  if  he  will  keep 
cler.rly  in  mind  the  nr.ture  of  the  problem  he  is  to  solve.  "He  is 
c.pt  to  entangle  himself  in  trying  to  remember  rules  and  methods 
by  which  to  reach  a  solution"",  (T'cGoodv/in) .  Statement  of  certain 
more  or  less  evident  corollaries  of  preceding  discussion. 

A  thorough  understanding  of  the  general  corollaries 
stated  below  will  be  of  inestimable  benefit  in  quick  and  ready 
analysis  of  problems. 

1.  All  straight  lines  and  planes  may  be  considered  as 
being  of  indefinite  extent.  Parts  not  of  such  lines 
and  plahes  lying  beyond  parts  having  actual 
existence  in  cases  considered  will  be  termed  " 
"imaginary".  .{Pig.  11). 

2.  (a)  A  point  which  is  not  in  light  cannot  cast  a 

real  shadow. 

(b).  Every  real  shadov/  line  must  cast  a  real  shadow 
and  this  real  shadow  cannot  lie  within  another 
real  shadow,  Host  of  the  blunders  in  casting 
shadows  are  due  to  a  neglect  ownunderstanding 
of  these  two  statements.  For  instance,  in  Fig. 
12  it  is  evident  that  point  "a"  cannot  cast  a 
shadov/  on  the  wall. 

3.  (a)  The  shadow  of  a  straight  line  on  a  plane  may 

be  determined  by  the  shadovj  of  any  two  of  its 
points  ofc  the  plane,  ^'^aturally  the  shadows  of 
the  points  at  the  ends  of  the  line  are  the 
ones  most  advantageous  to  find.- 

(b)  The  shadows  of  any  line  on  any  surface  may  be 
determined  by  finding  the  shadows  of  adjacent 
points  of  the  lines.  As  could  be  expected,  this 
method  is  a  very  cumbersome  one,  and  such 
shadows  will  be,  wherever  possible,  found  by 
less  lengthy  processes. 


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7. 

SHADES   AND   SHADOVfS 

13.  THE   GENERAL  METHOD. 

Sought  for  results  may  be  acheived  by  cumbersome  or  by 
easy  methods.  So  in  the  finding  of  shades  and  shadows  on  architect- 
ural drav/ings,  certain  methods  have  been  found  to  give  results  easil' 
and  accurately.  Those  general  methods  found  to  be  the  most  applicable 
are: 

1)  The  method  of  oblique  projections, 

2)  "^e  method  of  circujascribing  surfaces. 

3)  The  method  of  auxiliary  shadov;s, 

4)  The  slicing  method. 

Each  separate  method  ivill  be  discussed  in  detail  in  its 
proper  sequence,  and  with  its  proper  applications.  All  of  the 
methods  or  but  one  method  may  be  conveniently  used  in  the  casting 
of  shadows  on  any  one'  object.  The  use  of  a  little  common  sense  and 
visualizing  faculty  are  essential  ifi  the  student  is  to  do  any 
particular  probl"!u  problem  accurately  and  quickly. 

14.  THE  METHOD  OP  OBLIQUE  PROJI^CTION. 


The  method  of  oblique  projection  consists  simply  in 
drawing  on  the  projections  of  the  object  the  forty-five  degree  line 
representing  the  rays  tanrent  to  an  object  or  passing  through  itd 
shade  line  and  then  in  finding  the  points  ;There  the  rays  strike  any 
uther  object  involved  in  the  problem,  these  points  of  interesction 
giving  the  outline  of  the  shadov;. 

Thffis  method  is  simple  and  direct,  but  n.'/hrr'-.j  ly  can  be 
used  only  -when  the  plan  or  side  elevation  can  be  represented  by  a 
line.  Otherwise  it  is  impossible  to  find  cirectly  points  at  wHicE  rays 
s tr'ike  the  given  surface.  For  example,  in  plan  the  surface  of  a 
cylinder  with  vertical  elements  can  be  represented  by  a  circle  but 
the  surface  of  a  torus, scotia,  or  cone  cannot.  Hence  in  the  latter 
case.s  some  method  other  than  that  of  'direct  projection  must  be  used 
if  shades  or  shadovTS  are  to  be  found  on  these  surfaces.  Theoretically 
this  method  requires  the  finding  of  the  shadows  of  all  points  in  a 
line,  but  practically,  under  the  assumption  made  in  this  study-- 
namely;  rays  of  fixed  direction  parallel  v;ith  each  other--the 
shadov/s  of  certain  points  and  lines  on  certain  surfaces  in  certain 
portions  will  be  always  the  sr^.me.  They  maj  be  stated  as  follows: 

1)  The  elev'^tion  of  the  shadov/  of  a  point  on  a  vertical 
plane  v-iill  '\lways  lie  on  a  forty-five  degree  line  to  the  right 
of  the  elevation  of  the  point  in  front  of  the  plane.  (Fig.  13). 


8. 

SHADES  MID  SHADOWS 

2)  The  shadow  on  a  given  plane  of  any  line  which  is 
parallel  to  that  plane  is  a  line  equal  and  parallel  to  the  given 
line  and  lies  to  the  right  of  the  line  a  distance  equal  to  the 
distance  of  the  line  in  front  of  the  plane.  Fig.  14). 


parallel. 


3)  The  shadows  of  parallel  lines  in  any  plane  will  be 


4)  The  shadow  of  a  line  perpendicular  to  an  elevation 
plane  will  in  front  elevation  be  always  a  forty-five  degree  line  no 
matte'r  v/hat  be  the  fonii  or  position  of  the  objects  receiving  the 
shadow.  The  shadow  of  a  line  being  formed  by  the  intersection  of  a 
plane  of  rays, through  the  line  with  the  surface  considered,  in  this 
particular  case,  will  coincide,  of  course,  with  the  elevation  of  the 
plane  of  rays  this  plane  will,  therefore,  be  itself  perpendicular 

to  the  surface  receiving  the  shadow  and  will  appear  in  elevation  as  a 
forty-five  degree  line.  Eig.lS). 

5)  The  shadow  in  plane  of  line  perpendicular  to  the  plan 
plane  should  be  a  forty-five  degree  line.  The  reasoning  given  in 
(4)  should  be  sufficient  to  malte  the  point  clear.  Fig.  16). 

6)  The  shadow  of  a  vertical  |bine  on  an  inclined  plane 
v/hose  horizontal  lines  are  parallel  to  the  elevation  plane  is  an 

V inclined  plane  whose  slope  is  equal  to  that  of  the  given  plane. 
The  most  frequent  application  to  architectural  problems  is  found 
in  shadows  of  dormers  and  chimneys  on  roofs.  (Fig.  17), 

7)  The  shadow  of  a  vertical  line  on  a  series  of  horizont'  I. 

mouldings  is  equal  in  front  elevation  to  the  profile  of  the  right 

section  of  the  m.ouldings.  (Fig.  18).  The  shadow  line  of  course  moves 

to  the  right  as  the  contour  recedes.  This  shadow  of  such  -frequent  <> 

occurence  in  architectHral  problems  is  too  often   .        drawn. 

incorrctly 

8)  The  shadows  of  horizontal  :>lines  either  parallel  or 
perpendicular  to  the  elevation  plane,  on  a  vertical  plane  receding 
diagonally  to  the  left  at  an  angle  of  forty-five  degrees--are  of 
the  parallel  lines  forty-five  degree  lines,  sloping  dovmward  and 

to  the  left,  and  of  tjie  perpendicular  lines  forty-five  degree  lines 
sloping  down-ward  and  to  the  right,  (Fig.  19).  These  shadows  are 
often  used  in  finding  of  auxiliary  shadows. 


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SHADES  AND  SHADOWS 

9)  ^he  shdde  line  on  a  curved  surface  whose  elements  are 
horizontal  or  vertical  straiccht  lines,  is  found  by  drawing  the 
elevation  of  a  ray  tangent  to  the  profile  of  the  surface.  (Fig, 20), 
Evidently  the  shadoiv  cast  over  such  a  surface  "by   a  straight  line 
•vvhich  is  parallel  to  the  elements  of  the  surface  can  be  found  as 
shown.  (Fig.  21), 

With  the  knowledge,  therefore,  of  the  definite  positions 
that  the  shadows  of  these  lines  vjhich  form  boundary  lines  are  to 
practically  all  surfaces  appearing  in  architectural  vrork  take,  the 
finding  of  shadows  of  even  the  most  complicated  surfaces  becomes    / 
comparatively  easy.  Often  the  deter.nination  of  the  shadow  of  a      \ 
single  point  xvill  suffice  for  the  determinc?.tion  of  an  entire  gr^up  /  \ 
of  shadov/s.  An  attempt  must  always  be  made  to  use  as  little  as 
possible  either  plan  or  side  elevation  in  determining  shado-^firs-- 
for  they  are  often  in  architectural  vjork  at  different  scale  from 
the  front  elevations,  and  if  used  to  a  large  extent  would  have  to 
be  redra.wn--a  lengthy  and  entirely  uncalled  for  proceeding. 


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10. 

SHADES  Am   SHADOVJS 
Nomenclature 

For  the  purpose  of  clearness  in  reading  of  diagrams 
given,  the  followinj;;  nomenclature  has  been  adopted: 

R  -  Ray  of  lic'ht  in  space  at  Conventional  Angle. 

R^.-     Front  Elevation  of  Ray. 

R2-  Side  elevation  of  Ray. 

R3-  Plan  of  Ray. 

(|)  -  True  angle  of  Ray^' 
-:■  V  -  Vertical  plane  of  projection  or  Front  Elevation  Plan^ 

P  -  Profile  or  Side  Elevation  Plane. 

H  -  Horizontal  or  plan  plane. 

X  -  Any  other  plane. 
Let  A      Any  point  in  space. 
Then  Aj  -  Any  point' in  front  elevation. 

A2  -  Same  point  in  plan. 

A3  -  Same  point  in  side  eleVation. 
Let.  Als-  Front  elevation  of  shadov;  of  point  A.' 

A2S      Plane   of   shadovf  of  point  A. 

Ajg  Side  elevation  of  shadow  of  point  A. 

GL-  Ground  Line-Line  of  intersection  of  V  Plane  and 

H  plane. 

NOTE:  In  lettering  of  all  problem',  plates  this  nomenclature  is  to 
be  follo'vved. 

The  Architectural  terms,  front  elevation  (or  usually 
Elevation),  Side  Elevation,  and  Plan"^  are  to  be  used  in  preference 
to  the  terms  V  Projection,  P  projection,  and  H  projection. 


S'.i 


11. 

THE  SHADOVfS  OF  CIRCLES. 

1^.  SHADO'^S  OF  CIRCLES  IN  PLANES  PARALLEL 
TO  PLANE  TiECEIVING  SHADDW7 

It  is  quite  evident  that  the  shadov/  line  of,  for  instance, 
a  circular  flat  disl-:  on  a  plane  v/ill  be  formed  "bj   the  intersection 
of  a  cylinder  of  rays  v/ith  the  plane  in  question,  and  that  the 
intersection  of  the  cylinder  u-ill  be  a  circle  or  an  ellipse  depend- 
ing upon  v/hether  the  disk  vfere  in  a  plane  parallel  to  the  given 
plane  or  in  a  plane  at;  an  anjV.le  vvith  the  ^iveh  plane,  (Fig, 22). 
It  is  also  evident  that  in  the  first  instance,  the  shadow  line  will 
be  a  circle  of  exactly  the  same  radius  as  the  disk  casting  the 
shadovRji.  and  that,  therefore,  the  finding  of  the  shadov;  of  the  center 
of  the  circle  will  be  sufficient  to  determine  the  complete  shadow. 
The  arch  is  the  common  architectural  form  in  -which  circles  occur  in 
such  a  position. 

16.  SKADOITS  OK  VERTICAL  AND  HORIZONTAL  PLALIES . 

TiTien  the  shadow  line  is  an  ellipse,  by  methods  of  direct 
projection  from  plan  or  dide  elevation  as  auxiliaries  can  be  found 
a  nuinber  of  points  of  shadov-'s  of  the  circumference  of  the  circle. 
Usually  circular  forms  occur  in  architectural  work  in  planes 
perpendicular  to  the  vertical  plane  of  projection  and  parallel  to 
the  horizontal,  or  in  planes  perpendicular  to  both  planes  of  pro- 
jection, and  the  shadovj  of  such  circles  are  usually  cast  on  a 
vertical  or  elevation  plane,  though  sometimes  on  a  horizontal  or 
plan  plane.  Since  the  center  of  the  circle  is  a  point,  it  is  quite 
easy  to  determine  its  shadow,  which  determines  naturally  the  center 
of  the  ellipse ?shadow.  The  major  and  minor  axes  of  the  ellipse  of 
shado'w  are  then  determined. 

The  architect  must,  of  cours,  determine  if  possible  bjf 
methods  of  reasoning,  those  shado-w  points  that  "will  be  of  greatest 
importance,  and  through  them  must  construct  the  ellipse  of  shadow. 
The  simplest  and  most  accurate  method  of  determination  is  as  follows. 
(Fig.  23). 

The  shadows  of  the  circumscribed  and  the  inscribed  squares 
are  first  found,  using  the  already  found  shadow  of  the  center  of 
the  circleas  a  point  for  syrmnetrical  construction.  The  shadov/s  of 
the  median  and  diagonal  lines  are  easily  found.  The  points  at  \"jhich 
the  ellipse  of  sha.doi7  crosses  the  diagonals  is  found  as  shown  in  the 
figures.  The  tangents,  v/hich  of  course  are  parraia^el  to  the  diagonals. 
are  usually  drawn' to  serve  as  a  guide  in  freehand  construction  of  the 
ellipse  of  shadow.  Since  the  circle  is  a  contintkous  curve,  if  through 
any  inaccuracy  of  construction  points  of  shadow  found  do  not  give 
continuoTis  curve,  then  these  points  must  be  disregarded  and  the  curve 
dra;vn  through  the  greatest  n\imber  that  lie  on  a  continuous  curve. 


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12* 

SHADES  AND  SHADOWS. 

17.  SHADOWS  ON  INCLINED  PLANES . 

The  same  methods  of  reasoninrr,  as  used  for  shadows  of 
circles  on  vertical  and  horizciital  planes  give  the  construction 
of  the  shadovf  on  an  inclined  plane,  as  shown  in  Fig.  24, 

18.  SHADOWS  ON  45-^  AUXILIARY  PLANE. 

'^he  shadcvr  or.  a  vortical  plane  at  45  degrees  passing 
through  its  center,,  of  a  circle  in  a  horizontal  plane  perpendicular 
to  the  elevation  plane  is  a  circle  (Fig,  25), 

19.  CONCLUSION, 

It  T^Uf^t  above  all  -"Ise  be  remembered  that  the  shadow  of 
any  circle  must,  of  course,  be  completely  within  the  shadow  of  the 
circumscribed  square  and  will  be  tangent  to  that  shadow  at  points 
where  the  original  t:ircle  is  tangent  to  the  circumscribed  square, 
Time  honored  blunder 3  i"u  the  casting  of  the  eaadov/s  of  circles  may 
be  almost  entirely  avoided  by  the  accurate  finding  of  the  shadow 
of  the  circur!iEcriT;ed  square  even  though  the  inscribed  square  is  not 
found. 


13, 

20.  THE  SHADES  ON  MID  SHADQVfS  OF  SURFACES  OF  REVOLUTION. 

Surfaces  of  revolution  are  created  "by  revolving  lines 
straight  or  curved  or  both  about  a  fixed  axis  or  series  of  axes, 
.  In  the  forms  comraonly  met  v;ith  in  architectural  objects,  the 
surfaces  of  revolution  are  generally  either  vertical  or  horizontal, 
so  that  the  shapes  created  are  more  simple  to  deal  xvith  than  those 
created  v;hen  the  axes  are  inclined,  '^e  study,  therefore,  vill  be 
confined  to  right  cylinders,  cones,  spheres,  tori  and  sco$;ias.  The 
shades  on  and  shadoivs  of  certain  of  these  surfaces  can  easily  be 
found  by  an  application  of  some  one  or  all  of  the  methods  mentioned 
at  the  beginning  of  the  discussion. 

21.  THE  SHADES  ON  AN  UPRIGHT  CYLINDER.  (Fig.  26). 

It  is  quite  evident  that  the  surface  of  an  upright 
cylinder  can  be  represented  in  plan  by  a  circle,  and  the  surface 
of  a  horizontal  cylinder  in  side  elevation  by  a  circle.  Hence  the 
method  of  oblique  projection  can  be  applied  in  the  finding  of  the 
shades  and  the  shadovrs.  The  lines  of  on  the  cylinder  -aill  evidently 
be  determined  by  tv.'o  planes  of  rays  tangent  to  the  surface,  They  v/ill 
come  tangent  along  a  vertical  or  horizontal  element  of  the  cylinder 
and  can  be  represented  in  plan  or  side  elevation  as  the  case  may  be 
by  lines  tangent  to  the  plan  or  side  elevation  of  the  line  represent- 
ing the  surface  of  the  cylinder.  From  the  points  of  tangency  thus 
determined  are  secured  the  lines  of  shades.  One  line  of  shade  is  of 
course  invisible  in  the  front  elevation.  The  other  is  a  little  less 
than  1/6  of  the  v;idtheof  the  elevation  of  the  cylinder  to  the  left 
of  the  right  profile  of  the  cylinder.  This  proportion  is  a  con- 
venient one  to  remember, 

22.  THE  SHADOWS  OF  CYLIND~RS.  (Fig.  26). 

The  outline  of  the  shadow  of  a  cylindB±cal  surface  on  a 
plane  can  evidently  be  determined  by  finding  the  shadows  of  the  tvo 
shade  lines-~vjhich  are  straight  lines,  and  the  shadovjs  of  those 
portions  of  the  circular  outline  of  the  top  and  base  that  lie  betv.'een 
the  points  at  which  the  lines  of  shade  cross  the  bases.  These  will 
be  elliptical,  and  it  is  best  to  determine  by  methods  already  given 
for  the  shadows  of  entire  circles.  As  can  readily  be  seen,  the  ^■'.  ■-. ' 
width  of  the  elvation  of  the  shadovf  of  a  cylinder  on  a  vertical 
planerwill  be  equal  to  the  diagonal  of  a  square  having  the  diameter 
of  the  cylinder  for  a  side  and  -Till  also  be  symmetrical  about  the 
shadow  of  the  axis  of  the  cylinder.  This  fact  can  be  conveniently 
put  into  use  in  many  cases. 


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14. 

23.   THE  SHADOWS  ON  CYLII'DRIC.'.L  SURFACES. 

1.  The  Sh?.do-as  of  a  Straight  Line  on  an  Upright  Cylinder, 

The  shadov:  of  a  straight  line  parallel  to  both  the  V  and  H 
planes  nill  be  a  circle  v/hose  radius  is  equal  to  the  radius  of  the 
cylinder  and  the  elevation  of  vrhose  center  lies  on  the  elevation 
of  the  axis  of  the  cylinder,  belo:'.'  the  elevation  of  the  line  a 
distance  equal  to  the  distance  of  the  line  in  front  of  the  axis  of 
the  cylinder.  This  should  be  clear  from  Fig,  28. 

The  shadov;  on  the  surface  of  a  line  perpendicular  to  the 
elevation  plane  and  parallel  to  the  plan  plane  is  a  line  at  45/'/! 
(Art.  14,  paragraph  4).  The  shadow  of ' the  end  of  the  line  would  be 
found  by  direct  projection.  (Fig,  28).  The  shado'.v  of  any  other 
straight  line  would  be  formed  by  the  direct  projection  onto  the 
surface  of  the  cylinder  of  enough  points  of  shadov;  to  determine 
the  curve  of  shadov;. 

2.  The   shadov;  on  an  Upright  Cylinder  of  a  Larger  Cylinder 
whose  axis  coincides  v;ith  the  Axis  of  the  Smaller 
Cylinder. 

Evidentl-y  the  shadow  line  vrill  be  a  curve  rf  no  easily  constnur 
ed  geometrical  form.  Hence  it  becomes  necessary  to  find  of  that  curve 
by  means  of  direct  projection  enough  points  to  determine  the  directici 
of  the  curve.  Since  the  outline  of  the  shadov;  v;ill  be  determined 
by  the  shadov;  ofi  a  certain  portion  of  the  circle  bounding  the  lovjer 
surface  of  the  c^^linder,  by  means  of  direct  projection  from  points 
in  this  circle  on  the  sv\rface  can  be  found  any  nuraber  of  points 
desired.  However,  certain  points  are  of  more  importance  in  determin-t 
ing  shape  of  the  shadov;  than  others.  Those  points  are  naturally 
enough,  the  points  v;here  the  shadow  crosses  the  lighted  profile  of  the 
cylinder,  the  point  v;here  it  crosses  the  shade  line,  and  the  point 
vjhere   it  crosses  the  elevation  of  the  axis,  and  v;here  it  is  closest 
to  the  elevation  of  the  line  casting  the  shadov;,  3y  inspection  it 
ca.n  be  seen  that  the  highest  point  of  shadov;  lies  on  the  diagobai 
axis  en  the  left  of  the  center,  for  on  that  line  the  rays  strike  the 


:F 


surface  at  the  true  angle, --(The  angle  "p  ). 

Therefore,  it  is  determined  first  of  all  vrhat  points  on  the 
circle  cast  shadov;  on  these  lines  mentioned  and  thenas  many 
additional  points  are  determined  as  are  deemed  necessary  to  the 
correct  drawing  of  the  shadow.  If  the  object  be  small,  then  n?.tviu:-lly 
the  determination  of  the  four  points  mentioned  is  sufficient:  if  the 
object  be  large,  then  more  points  must  be  determined.  The  method 
of  determination  is  shown  in  Fig»  29,  The  p©int  on  the  profile  line 


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15. 

and  that  on  the  axis  of  the  cylinder  are  at  the  same  distance  beloiv 
the  line  casting  the  shad  or;.  The  determination,  therefore,  of  the 
position  of  either  one  v/ill  be  sufficient  to  determine  the  position, 
of  the  other. 

24.  THE  SHADES  AND  SHADOWS  OF  HOLLQH  CYLINDERS. 

In  the  d  ra;7ing  of  the  sections  of  buildings  it  is  often  neces- 
sary to  determine  the  shades  and  shadovrs  of  hollov/  cylinders,  as  for 
instance  in  the  section  of  the  cupola  of  a  dome,  or  that  of  a  hori- 
zontal barrel  vault  or  that  of  an  arch.  The  methods  of  determining 
the  points  necessary  to  give  the  correct  general  shapes  of  the  shades 
and  shadov;s  on  such  cylindrical  surfaces  are  shown  in  Fig. 30, 31, 3  2, 

25.  TIffi   lETHOD   OF  AUXILIARY  SHADOWS 

The  principles  upon  -which  the  application  of  this  method  is 
based  are :- 

1.  If  upon  any  surface  of  revolution  a  series  of  auxiliary 
curves  be  drawn,  the  shadov;  of  the  surface  will  include 
the  shadovjs  of  all  the  au::iliaries,  and  v;ill  be  tangent 
to  those  that  cross  the  shade  line  of  the  surface  at 
points  vjhich  are  the  shadows  of  the  points  of  crossing, 

2.  The  point  of  intersection  of  tvjo  shadov/  lines  is  the  shad  - 
ow  of  the  point  of  intersection  of  these  lines  if  they  are 
intersecting  lines;  if  the  shadow  of  the  point  where  the 
shadow  of  one  of  the  lines  crosses  the  other  line,  if  they 
are  not  interecting  lines.   If,  therefore,  the  shadows  of 
tvJO  lines  intersect  or  come  tangent  in  a  point,  the  position 
of  the  point  of  tangency  or  the  point  of  intersection  of 
the  tvjo  lines  casting  the  shadow  may  be  determined  by  trac- 
ing back  along  the  ray  to  the  line  in  question.  Evidently 
ease  of  construction  and  the  necessity  of  accurate  drawing 
malce  it  necessary  to  choose  auxiliary  lines  -whose  shadows 
may  be  easily  and  accurately  determined. 

26.'   THE  SHADES  ON  AND  SHADOVIS  OF  CONES. 

The  Conical  forms  ordinarily  met  with  in  architectural  work 
are  those  -with  vertical  axis--as  for  example,  the  roof  of  a  cylind- 
rical tower,  the  lower  part  of  a  wall  lamp,  etc.  The  d  iscussion  to 
follow  -will  be  limited,  therefore,  to  upright  cones. 


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-      'vVti 

■'■■-■    r  I        r'i  ~ 


■^■^    V*r:, 
*  -^^  ■'■     ,  1  i. 


>^:J 


'n    ,;,,     v-s, 


T'.,;     ■   V        -ifr,    I 


■f.t 


"'.f*'-    -     ;};^',;  . 


'   I. 


*  it      .  ; 


'•i'JOj 


16. 

The  surface  or  c.   cone  cannot  be  represented  in  any  one  of  the 
three  planes  of  projection  "b;'  lines.  Therefore,  the  shade  lines 
cannot  bo  found  by  the  processes  applied  to  the  cylinder.  Prom 
inspection  it  can  be  seen  that  the  sha,de  lines  v/ill  be  forned  by 
planes  of  rays  tangent  to  the  surface  of  the  cone.  Tliey  will  be 
strai^jht  lines  passing  ;^hrou£h  the  apex  and  crossing  the  line  of 
the  base.  Since  in  projection,  the  shade  of  the  apex  vrill  coincide 
v/ith  the  projections  of  the  apex,  then,  in  order  to  find  the  pro- 
jection of  the  shade  lines  it"  becomes  necessary  to  determine  only 
poj,nts  at  ;7hich  the  shade  lines 

cross  the  projection  of  the  base.  To  secure  those  points,  the 
method  -of  auxiliary  shado•■^rs  is  used. 

The  outline  shado"  of  the  cone  on  any  plane  v;ill  be  determined 
by  shc.doi^s  of  the  base,  the  apex  ?\nd   the  shade  line.  If,  therefore, 
as  in  Pia.  33,  the  shadovr  of  the  apex  and  the  shaov;  of  the  base  be 
cast  independently  upon  any  given  plane, 'then  the  shadov/s  of  the 
shade  lines  riust  he   Als,Bls,  -.nd  Als,Cls.  t'oints  3Is,  and  Cls, 
com?aon  to  the  shadovrs  of  the  shade  line  and  the  shadow  of  the  ba,se, 
nwxstl^9  the  shado'.vs  of  points  coTL--on  to  the  shade  line  and  base, 
namely, the  shadows  of  points  common  to  the  shade  where  the  shade 
lines  cross  the  base  line.  3y  passing  back  from  3ig^  and  Cig  along 
rays  of  light  to  the  base  ftin^be  found  points  D  and  C>  and  thas 
shade  lines  A  h   and  A  C  constructed. 

In  finding  the  points  3  and  C  geometrically  some  method  that 
■'/ill  give  absolute  accuracy  must  be  used.  Therefore,  to  find  the 
points  3  and  C  in  elevation  the  shado'vs  of  the  cone  on  either  a 
horizontal  plane  thron'^h  Its  base,  or  on  a  45  degree  vertical  plane 
through  its  a;:is  arc  used  as  auxiliaries.  Figs.  35  a.nd  36  illustrate; 
the  application  of  the  method  vfherein  is  used  as  an  auxiliary,  the 
shadov,'  on  the  horizontal  plane.  (Pig.  35)  In  the  geometrical  con- 
struction shovna  in  Fig,  36,  the  shadow  of  the  cone  Bg^*  -^29'  ^2,,  i^ 
simply,  for  the  sake  of  convenience  superposed  6n  the  elevation  of 
the  cone  and  points  Bl  and  Cx  thus  deter:;iined. 

In  Fig.  38 J  the  points  Ei  and  Pi  are  first  determined  by  cast- 
ing the  shadovr  of  the  cone  oh  the  45  degree  auxiliary  plane,  then 
since  the  shadovf  A1,E1  III  PI  is  that  of  the  base  of  the  cone  on  the 
45/-  auxiliary  plane,  then  from  El  and  PI  in  elevation,  rays  are 
passed  at  45',..  to  cut  the  elevation  of  the  base,  p.nCi   thereby  determine 
the  position  of  the  points  3i  and  Ci.  Tlie  shade  lines  AiBl  and  AiC]_ 
can  then  be  drav/n. 

27.   COKES  ^"ITHOUT  V-ISI3LE  SIlADE  LIUES. 

If,  as  in  Pig,  39,  the  profile  lines  of  the  cone  make  v/iththe 
hoiPizontal  an  angle  of  45%,  then  there  is  in  front  elevation  no 
visible  shade  line.  Tiie  shade  lines  in  plan  are,  however,  as  sho".vn. 
If, as  ■  x     *J'''.  in  Fig,  40,  the  profile  lines  make  an  angle  of  <t> 
or  less  tha  A  (J)  -vith  the  horizontal,  then  the  cone  has  no  shade 
line  in  plaiior  elevation. 


I  f 


.,.,.  p 


17. 


28.  THE  SHADOVfS  OF  CONES. 


The  outline  of  the  shadow  on  any  surface  can  be  secured  by  cas 
onto  that  surface  the  shadows  of  the  apex,  of  the  shade  lines  am 
the  profile  of  the  base.  If  the  surface  on  which  the  shadow  fallf^ 
a  plane,  the  shadow  of  the  axis  of  the  cone  should  be  first  detei - 
mined  I  then  on  the  shadow   line  can  be  determined  the  shadow  of  the 
apex  then  the  shadow  of  the  base.  Through  the  shadow  of  the  apex 
straight  lines  drawn  tangent  to  the  shadow  of  the  base  will  complete 
the  shadow  of  the  cone. 


xn£ 

of 

be 


29.  THE  SHADES  AND  SHADOVfS  OF  SPHERES. 

The  shade  lines  of  a  s^ihere  is  evidently  a  great  circle  of  the 
sphere  and  is  symmetrical  about  the  two  forty-five  degree  axes  in 
the  plane  and  in  elevation.  It  is  readily  understood  that ' the  point 
vvhere  the  shade  lineof  any  double  curved  surface  of  revolution  toucho? 
the  contour  lines  of  the  surface  in  plan  or  elevation  is  of   course  t|, 
be  found  at  the  point  of  the  contour  at  which  the  plan  or  elevation 
of  a  ray  is  tangent  to  it. 


Therefore,  Eig.41b,  to  find  on  the  plan  the  points  where  the 
shade  lines  come  tangent  to  the  plan  of  the  contour  draw  the  rays  R2, 
tangent  to  the  plan  of  the  sphere.  The  points  thus  found  evidently 
give,  (Fig, 41a),  on  the  elevation  of  the  equator  the  points  of  shade 
Ci&Di,  and  by  symmetry  the  points  Ei  and  Fi-  Points  Ai  and  B]_  are 
determined  by  drawing  rays  HI  tangent  to  the  elevation  of  the  sphere. 
Thus  are  determ.ined  six  points  on  the  shade  line.  To  determine  the 
points  xhich  will  give  the  length  of  the  minor  axis  of  the  ellipse, 
two  equilateral  triangles  with  their  apexes  at  Al  and  31  are  construct 
ed,  and  points  Gl  and  HI  thus  determined.  Through  these  eight  points 
the  ellipse  of  shade  can  then  be  drawn.  The  short  construction  for 
the  points  requil;'ed  is  shov/n  in  Pig,  41c.  The  outline  of  the  shadow 
of  the  ellipse  on  any  surface  will  'be  the  shadow  of  the  shade  line 
of  the  sphere,  and  sufficient  points  could  be  determined  by  direct 
projection  from  plan  and  elevation  to  fix  the  shape  of  the  curve, 
The  shadow  on  a  vertical  or  horizontal  plane,  hov/ever,  is  generally 
found  by  means  of  the  construction  shown  in  Fig.  41d,  and  is 
sufficiently  accurate  provided  the  shape  of  the  ellipse  of  shadow 
is  drawn  approximately  as  shown  on  the  figure. 

Many  mistakes  are  made  by  beginners  in  determining  for  instance 
the  line  of  shade  on  a  dome,  v/hich  is  of  course  only  part  of  a  sphere. 
The  visible  shade  line  can  be  determined  accurately  only  by  completing- 
the  snhere. 


. . '  ■; 


T:/. 


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18, 

30.  raE  SLICING  HETHOD. 

In  this  method  the  object  casting  shadovr  and  the  object  receiv- 
ing the  shadov/  are  first  cut  through  b^'  a  vertical  planes  pari^allel 
to  the  rays  of  the  light.  The  elevations  of  the  lines  of  intersection 
of  the  planes  v;ith  the'  surfaces  in  question  are  then  determined  and 
then  points  of  shade  and  shadoxv  are  determined  on  these  slices. 

The  process  is  explained  by  Pig,  42,  the  plan  of  the  obj'^ct 
is  shovrn  at  (a),  Tlie  plan  of  a  vertical  plane  at  45;/!'  passed  through 
the  object  nould  be  represented  by  the  line  (1).  The  elevation  of 
the  line  of  intersection  of  that  plane  viith   the  surface  given  would 
be  the  dotted  line  (l!*,  Tlie  shadow  on  the  scotia  ?70uld  be  cast  by  thp 
circular  edge  x  y.  The  shadow  of  point  Bl  vfould  evidently  lie  at  the 
point  Bis  '-fbere  the  elevation  of  ray  through  (3i)  strikes  the  line 
of  intersection  (1).  Plane  (2)  vfould  give  intersection  (2),  and  the 
elevation  of  a  ray  through  CI  v;ould  give  the  shadovf  C]_g,  By  thB 
use  of  a  sufficiently  large  number  of  accurately  constructed  slices 
enough  points  of  shadow  ■m,?.y  be  determined  to  fix  the  curve  of  the  r. 
shadov/  line. 

Because  it  is  a  quite  easily  understood,  this  method  of  determn  - 
ing  shade  and  shadov.r  is  apt  to  be  abused  by  the  beginner.  Firstly,  i .. 
must  be  distinctly  remembered  that  the  accurate  construction  of  each 
slice  is  ",  slow,  tedious  process  and  that  the  line  of.  slicing  when 
determined  v/ill  be  sufficient  to  give  but  one  point  of  shade  or 
shadow  and,  secondly,  that  a  point  of  shade  determined  by  drawing  a 
ray  tangent  to  such  a  slice  line  v;ill  not  be  accurate  in  its  position. 
Vihile  the  position  of  the  slicing  planes  may  be  chosen  so  as  to  give 
the  most  easy  construction  and  valuable  results,  yet  the  slicing 
method  is  one  to  be  used  only  v/hen  noj.  other  xzan  be  supplied.  It 
cannot  be  used  '^.t  .small  scale.  The  shapes  of  shadow  and  shade  lines 
determined  by  this  method  must  be  so  determined  at  a  scale  that  will 
allov;  of  accurate  construction  and  -nay  then  be  copied  at  small  scale 
by  proportion. 

3y  means  of  this  method  c"n  be  determined  the  shades  and  shadows 
of  scotias,  and  those  on  the  more  co"nplicated  vase  forms.  However, 
those  shadov/s  en   often  be  m.ore '  accurately  determined  by  the  use 
of  the  45degree  auxiliary  plane. 

31.  THE   SHADOF   OP  A    CIRCULAR  IIICIIE 

v:iT:ii:^  &  spherical  head 


The  outline  of  the  shadow  will  be  formed  by  the  shadow  of  the 
circular  outline  of  th.e  head  of  the  niche  -^.nc.  by'  the  shadov;  of  the 
strai.  ht  li:..:..'  •■'■<:.-.h   rsp/'csents  the  left  hand  side  of  the  niche.  Tlie 
lines  casting  the  shadow  can  be  represented  in  plan  by  the  line 


:•  f 


C  r 


' ,       c  ■   ", 


■  '■  ■■1' 


^        -rx 


■  •   f  : 


-r  I   I   ! 


r^r:.''    ■■•• 


'oo:  ■  - 


•'i  J 


'1.1 1..-, 
r      -■  i  ■  ■   '  ■  ;■(   ■  h: 

.:vi:  ,   0.' 

J I 
,"■      ' :'.  ■i~ 


;    (■ 


=:!:•.(:  ^ 


r  ■ 


- .t  i  '  ■  r.  .:)'-^       J-' 


i    ::^0.^/ 


'■:0    rf" 


;•  ;(  ■'. 


'  o '■!■-•  ■       ■  ■' 

^.    .-f    ...   ,      ,-.    r^  ■ 

.t  ».        ■   ■  •      ■     *■ 


■:r.    :^o  ..•  ■ 

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J. 


r 


M-r,-    vi^. 


.    X  ^  -•'       


19.   . 

A2  C2  B2,  Fig.  4  6.  The  Portion  of  the  niche  in  elevation  up  to 
line  Al  01  Bl  can  be  represented  in  plan  by  a  semicircle,  i-'oints  of 
sbadov;  Als  Els  Dls  can  be  found  by  direct  projection  from  the  plan 
and  elevation.  The  curve  of  shadov;  v;ill  evidently  start  at  Als  and 
end  at  Gls  vrhere  the  ray  R]_  comes  tangent  to  the  elevation  of  the 
profile.  To  find  points  of  shadov/  on  the  spherical  head  of  the  niche 
auxiliary  vertical  planes  such  as  x   and  y  are  passed  through  the 
surface  of  the  niche  and  the  elevation  of  the  lines  of  their  inter- 
section viith  the  surface  of  the  niche  found.  On  these  planes  are 
then  cast  the  shadow  of  the  semicircular  outline  of  the  head  of  the 
niche  and  the  points  Fls  and  Gls,  thus  Pis  and  Gls  are  points  of 
shadov/  of  the  semicircular  head  <£    the  niche  because  they  are  points 
common  to  the  surface  in  question  and  to  the  shadov;  of  the  semicir- 
cular head. 

At  small  scale  the  points  of  shadov/  to  be  determ.ined  are  /Is, 
Dls,  and  Cls.  Als  and  Cls  are  easily  determined,  and  Dls  may  be 
placed  on  the  elevation  of  a  ray  through  On  a  distance  from  01 
equal  to  l/3  of  the  radius  of  the  niche.  (Pig,47).  Tne   shadov/  line 
must  not  cross  the  outline  at  Cls  but  must  be  tangent  to  the  outline 
at  that  point. 

32.  THE   SHADES    ON  AW   THE   SHADOV'S    OP   HORIZONTAL  TORI. 

As  in  the  case  v/ith  all  double  curved  surfaces  of  revolution, 
the  shade  line  v/ill  be  symmetrical  in  plan  about  the  two  45  degree 
axes.  If  there  can  be  d  etermined  accurately  several  points  on  that 
±Lade  line,  then  same  can  be  drav/n  v/ith  accuracy  through  the  points 
thus  determined. 

33.  THE  5ADE  LINE  ON  i  TORUS . 

From  the  theorem  that  the  point  v/here  the  §i  ade  line  of  any 
d  ouble  curved  surface  of  revolution  touches  the  contour  line  of  the 
surface  in  plan  or  elevation,  is  found  at  the  point  of  the  contour 
at  v/hich  the  plan  or  elevation  of  a  ray  is  tangent  to  the  contour, 
there  can  be  determined   the  position  in  elevation  cf  the  'ooints  El 
Al  Pig.  4  3,  By  symmetrical  construction  can  be  found  the  ooints 
CI,  HI.  Prom  the  plan  bj;  drav/ing  rays  tangent  to  the  plan  of  the 
contour,  vhich  is  the  equatorial  circle  of  the  torus,  can  be  located 
points  Gl  and  Dl.  If  nov/,  there  c^n  be  c etermined  the  hir'.ir.st  znd 
lov/est  points  of  ±i  ade  then  the  shade  line  can  be  constructed  -/ith 
sufficient  accuracy ,  It  is  evident  that  the  lovest  and  the  hi^hect 
points  v/  ill  lie  in  plan  somev/here  on  the  axis  of  the  torus  that  is 
parallel  to  the  rays  of  light  and  that  point  v/ill  be  v/here  the  ray 
cf  light  stri>:es  the  surface  of  the  torus  at  the  angle  0.  If  the 
torus  be  then  revolved  about  its  vertical  axis  unt?.l  the  point 
comes  into  the  vertical  plane  of  projection  then  can  be  determined 
the  horizontal  element  on  "vhich  the  point  lie  s.  It  lies  on  the 
45  degree  axis  in  plan.   3y  finding  the  plan  or 


+    ?    ' 


■■T .'.  ■ 


.,  f 


•  r' "      f. 
;l<.v    _  ■• 


1     •       ~l   ■.  .■'     -' 


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■.>.!■••■ 


.'  i  ■ 


■'•■  3 


f*V 


V  ■■ 


Ai 


20. 

horizontal  elements  it  deternines  the  position  of  the  points 
desired,  namely,  Bl  and  Fl.  The  construction  is  shown  in  Fig.  43 

34.   ^tie  SHADprr  OF  A  TORUS. 

The  outline  of  the  shadov;  of  a  torus  is  the  shadow  of  its 
shade  line,  and  can  be  cast  on  any  surface  by  the  direct  projection 
of  points  in  plan  and  elevation.  Hov;ever,  as  shov/n  in  Fig.  44,  the 
shador  line  so  nearly  corresponds  to  shadov/  of  the  equatorial  circle 
of  the  torus  that  at  small  scale  the  shadovj  of  the  equatorial  circle 
can  be  used  as  the  shadov/  of  the  torus. 

55.   THT  OVAL  CURVE  _0F  THE  TORUS . 

The  Shadovr  of  a  torus  on  the  45  degree  auxiliary  plane  is 
found  by  casting  on  that  plane  the  shadovjs  of  various  horizontal 
elements  of  the  torus  and  dravjing  the  envelope  of  those  shadows.  The 
construction  is  shovm  in  Fig.  45.  This  ±i  ad  ovx  is  used  frequently  as 
an  auxiliary,  particularly  in  the  find  ing  of  the  shadov/s  of  straight 
lines  and  circles  on  the  surface  of  tori. 


.'],T     'Vr, 


r    ■        rs' 


rx 


'So-i., 


■'  '<■  y  'r 


r:.  r~ 


::r^r.,-..- 


'•* '•'^-   '-  ?i' 


THE:  "Rays 
Op  Light. 


YlGUi;>£:  Z 
ThEtCdmvention 
PlKECTION 

It  If:  Ray  ^ 

LlCMT 


6HADKand5HAD0W5 
PLATt  I. 


L=  Lateral  Rav^^. 

T=TANCEMy    f^AY& 

T  =  Incident  Ray5 


LE-TABCDH-IGHBEACUBL 
AG--PlAGDNALOfCuBIL- 

R=  Ray  0f  Light  Paralul 

To  Direction  c7Piagonal 
I^rPROJECTioN  Of  Ray  On  V 

Rj\NL  Or  Elevation  Or  \^av. 
R/P^ojictionOfRavOu  H 

Flake  Or  Plan  Of  ]^AY 
1^=  PROJECTION  Of  Pay  On  P 

FUkie0r5ideIllvatio\^0f!^,y 
/  AGC  True  AncleOpRa^ 

Oi^  Angle 'I'. 


Geometric  CoN5TeucTioN  Fo^  Angl£<^ 


F\GUR.&  5 
ThErTHREtPROJECTIONS  OfRaY  R. 


Side 


fpONT 


Plan 


If 


FiGURI:  4. 
PLANE  0PRAY5- 
.  DEinNiyioN! 


5HADI:5  8  SHADOWS 


FiGURfc    5- 
PO\NT<5  OMOSS- 
R)iNTp  a  b.C.d  ore-  POINTO   Of  L055. 


NVI5IBLE  Shadow 


^'*^  V15IBLE  Shadow 


TlGURt  6 . 

lNVI5lBLt  AND  VISIBLE  SHADOWS. 
DtFiNlJION" 


FIGURE:  7- 


flGUR.E:  8 
INX  SHADOW  Of  ABFALL5  ON  PLAMt 
IN  "5  SHADOW  0f-AB?ALL5  PARTLY  OM 
GROUND  .PARTLY  ON  CURVEP  5URFAC& « 


Figure  9 


6HADE5  ^  SHADOW. 
Plate  IE 


DXYC  15  AN  Imaginary  T 

Of  THtTOPO?THt  box-  y ,  i  .^  -^i. 

Imagina^^Y  IinE:-Thl5hadgw  0? 

THE  Cube  May  BE  BEi'Ond  LINE  LK 

AND  THE:  PORTION  mOND  WOULD 

BEIMACINARV. 


\     \ 


flCUR.E  12 


•  •^ 


FlGUl^tlO 


<5hADL§?< -Shadows 

Plate  E 

THE.6hAD0v/5  Of 
Bl)    Jtf  Toyi/the 


PhieSPECTIVEr  SKETCH 
APPOINT  lN5i^ACL 


IhE  Shadow  Of  Anv  Point 


f  IGUCt  14 
Shadows  Of  Lines  P.aralell  TctL£vATioNR.ANt 


F)GUR.t  15 

Shadow  Of  A  Imt  PECPtNOicuLAR  TotuEVATioN  Plane. 


i 


6HADE5  &  5HAP0W5 
PlateV 

p3iDt  Elevation 


F"TJ15P  16 


ihE-OHAD0Vv5  Uf-M  LiNL  I^El^PtNDICULAli'IbPLAN 

Plane  On  PLAN?LANt, 


A^ 

As' 

\ 

-> 

/ 

/ 

K 

i^ 

^35 


Figure  17 

Shadow  Of  VEBTicALLiNt  On  iNCUNLDR-ANfc 


1 I M 


S-;^---^-Ki^^ 


%  -^- 


\ 


te- 


:^ 


PT 


^ 


az". 


?IGUl^El8 

Shadow  Of  Vleticai-  Line  On  Cecils 

Of  HORlZOtHyAl-    M0UL-DINC5. 


i 


X 


CHADE5?)f5HAD0W5 

Plate  ?I 

5ij   J  ft- Fo3<fythe 


X. 


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Figure  I9fd) 

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Q516  61L6S  COOl 

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3  0112  089506593 


